Problem:
Let z=a+bi be the complex number with ∣z∣=5 and b>0 such that the distance between (1+2i)z3 and z5 is maximized, and let z4=c+di. Find c+d.
Solution:
The number z maximizes ∣∣∣​(1+2i)z3−z5∣∣∣​=∣z∣3⋅∣∣∣​1+2i−z2∣∣∣​. Because ∣z∣3=125 is fixed, it follows that z2 points in the direction of −1−2i with length 25, and thus z2=−δ(1+2i) for some positive real number δ. Thus 25=δ∣1+2i∣=δ5​ and δ2=125. Finally, z4=125(1+2i)2=125(−3+4i), and the requested sum is −375+500=125​.